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As always, Hall is an invaluable reference, particularly the chapter on Time, Dose and Fractionation in Radiotherapy (Chapter 22). This entry is mainly based on that chapter.

Overall Treatment Time and Dose

The first observed effects with regards to time and dose were that as treatment time increased, the dose required to achieve a similar effect (the isoeffective dose also increased.

Isoeffect Curves

Isoeffect curves were first published in 1944 by Strandquist. They showed the effect of treatment time on the the dose required for certain effects.

Nominal Standard Dose

The nominal standard dose was the first widely used equation developed to determine biological effect based on the number of fractions and the overall treatment time. It was developed by Ellis in the 1960s from data on skin reactions, and was frequently used until supplanted by the more accurate linear quadratic and alpha beta models. It is given by the following formula, where T is treatment time in days and N is the number of fractions.

The NSD (nominal standard dose) constant represented the tolerance of skin. It was also applied to other organs and connective tissue. If the calculated NSD value for a fractionated treatment was less than the tolerance NSD value, then it would be 'safe' to increase the amount of treatment until tolerance was reached.

Partial Tolerance (PT)

The partial tolerance was introduced by Ellis to counter the inability of the NSD model to adjust for changes in treatment regimen (from 5 to 3 fractions per week) or for gaps in treatment. The partial tolerance of a treatment course x (PTx) was given by:

(3)

\begin{align} PT_x=\frac{N_x}{N_{T_x}}.\text{NSD} \end{align}

Where Nx represented the number of fractions given in a treatment, and NT was the number of fractions required to reach the tolerance of the tissue. The NSD value represented the tolerance of the tissue in general.

The various partial tolerances of a treatment could then be added up:

(4)

\begin{equation} PT_{tot}=PT_1 + PT_2 + ... + PT_n \end{equation}

It was also realised that gaps in treatment increased the tolerance of tissues. This was accounted for by another equation which used the rest period R to adjust the partial tolerance:

Problems with the NSD model

I haven't finished going into the NSD model, but as you can see it becomes very complicated with altered treatment regimes. It was also inaccurate due to reliance on acute skin toxicity data. Finally, it also assumes repopulation is linear, whereas human data shows repopulation does not become a significant factor for about 4 weeks. It was also very cumbersome and difficult to use when treatments became more complicated.

Cumulative Radiation Effect (CRE)

The Cumulative Radiation Effect was proposed by Kirk et al in 1971. It was an improvement on the NSD, by substituting factors related to dose per fraction ($d=\frac{D}{N}$) and interval per fraction ($x=\frac{T}{N}$). The subsequent equation eliminated the total dose (D) and treatment time (T).

(6)

\begin{align} CRE = \frac{dN^{0.65}}{x^{0.11}} \end{align}

The CRE functioned in a similar way to the NSD - if the calculated CRE for a treatment was less than the tolerance CRE, then the treatment would be 'safe'. If different treatment schedules were used, the CRE values could be added to give the total CRE for a treatment. Unlike the NSD model, the CRE model took variations in treatment schedule into account and did not require the partial tolerance concept.

The CRE model suffered from the same problems as the NSD with regards to reliance on skin toxicity and problems with repopulation calculations. It would be overturned by the linear quadratic equation and alpha/beta models.

Proliferation of Normal Tissues

The two major problems with the early formulae such as the NSD and CRE were:

Proliferation of normal tissues does not occur linearly as time increases

Early responding and late responding tissues undergo proliferation at different times

Sigmoid Increase in Normal Tissue Proliferation

Studies on mouse skin have shown that proliferation does not occur until about 12 days after starting fractionation. Beyond this point, it rises rapidly, requiring an extra 1.3 Gy per day to reach the same biological effect.
This sigmoid curve explains why the NSD and CRE models are inaccurate, as they assume the above line to be linear.

Differences in tissue types

Different tissues proliferate at different times after exposure to ionising radiation.
Late responding tissues tend to proliferate much later following treatment, usually after therapy has ceased, whereas early reacting tissues (and some tumours) begin proliferating during the course of treatment.
Hall sums this up well:

Prolonging overall treatment time, within the normal radiotherapy range:

Significantly reduces early treatment related effects

Has minimal effect on late treatment effects

Summary of Overall Treatment Time

Treatment time was observed by Strandquist in the 1940s to increase the dose required for the same effect

Initial modeling of this effect was done with NSD and CER equations, but these were inaccurate due to insufficient data to model them on

Early reacting tissues begin proliferation after a short time interval. This means that overall treatment time has a major effect on the effects seen.

Late reacting tissues begin proliferation after treatment has been completed (for most treatments). This means that overall treatment time has minimal effects on the effects seen.